where \(c_{i,k}\)  is the \(k\) -th dimension of the discrete code representation for item \(i\) . To learn these PQ centroid embeddings (i.e., \(\lbrace {a}_{k,j}\rbrace \) ), we adopt a popular method optimized product quantization (OPQ) [1]}, and then optimize centroid embeddings based on the text encodings of items from all the training domains. After centroid embedding learning, we can perform the index assignment (Eq. (REF )) independently for each dimension, and obtain the discrete code representation \({c}_i = \left(c_{i,1},\ldots , c_{i,D} \right)\)  for item \(i\) .
